Int-Alg Chapter 5 Summary and Review

Section 5.7 Chapter 5 Summary and Review

Subsection 5.7.1 Glossary

function
input variable
output variable
cube root
absolute value
proportional
direct variation
inverse variation
constant of variation
concavity
scaling
horizontal asymptote
vertical asymptote

🔗

Subsection 5.7.2 Key Concepts

A function can be described in words, by a table, by a graph, or by an equation.
Function Notation.

🔗
Finding the value of the output variable that corresponds to a particular value of the input variable is called evaluating the function.
The point $(a, b)$ lies on the graph of the function $f$ if and only if $f (a) = b .$
Each point on the graph of the function $f$ has coordinates $(x, f (x))$ for some value of $x .$
The Vertical Line Test.

A graph represents a function if and only if every vertical line intersects the graph in at most one point.
🔗
🔗
We can use a graphical technique to solve equations and inequalities.
$b$ is the cube root of $a$ if $b$ cubed equals $a .$ In symbols, we write
$b = \sqrt[3]{a} if b^{3} = a$
Absolute Value.

The absolute value of $x$ is defined by
🔗

$| x | = {\begin{cases} x & if x \geq 0 \\ - x & if x < 0 \end{cases}$

🔗
🔗
Absolute value bars act like grouping devices in the order of operations: you should complete any operations that appear inside absolute value bars before you compute the absolute value.
The maximum or minimum of a quadratic function occurs at the vertex.
Eight basic functions and their graphs are important in applications:

$\begin{matrix} f (x) = x f (x) = | x | f (x) = x^{2} f (x) = x^{3} \\ f (x) = \sqrt{x} f (x) = \sqrt[3]{x} f (x) = \frac{1}{x} f (x) = \frac{1}{x^{2}} \end{matrix}$
Two variables are directly proportional if the ratios of their corresponding values are always equal.
Direct Variation.

$y$ varies directly with $x$ if
🔗

$y = k x$

where $k$ is a positive constant called the constant of variation.
🔗

🔗
🔗
Direct variation defines a linear function of the form
$y = f (x) = k x$
The positive constant $k$ in the equation $y = k x$ is just the slope of the graph.
Direct variation has the following scaling property: increasing $x$ by any factor causes $y$ to increase by the same factor.
Direct Variation with a Power.

🔗

$y$ varies directly with a power of $x$ if
🔗

$y = k x^{n}$

where $k$ and $n$ are positive constants.
🔗

🔗
🔗
If the ratio $\frac{y}{x^{n}}$ is constant, then $y$ varies directly with $x^{n} .$
Inverse Variation.

$y$ varies inversely with $x$ if
🔗

$y = \frac{k}{x}, x \neq 0$

where $k$ is a positive constant.
🔗

🔗
🔗
Inverse Variation with a Power.

$y$ varies inversely with $x^{n}$ if
🔗

$y = \frac{k}{x^{n}}, x \neq 0$

where $k$ and $n$ are positive constants.
🔗

🔗
🔗
If the product $y x^{n}$ is constant and $n$ is positive, then $y$ varies inversely with $x^{n} .$
A graph that bends upward is called concave up, and one that bends downward is concave down.

🔗

Exercises 5.7.3 Chapter 5 Review Problems

Exercise Group.

Which of the tables in Problems 1–4 describe functions? Why or why not?

🔗

1.

$x$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$y$	$6$	$0$	$1$	$2$	$6$	$8$

🔗

2.

$p$	$3$	$- 3$	$2$	$- 2$	$- 2$	$0$
$q$	$2$	$- 1$	$4$	$- 4$	$3$	$0$

🔗

3.

Student	Score on IQ test	Score on SAT test
(A)	$118$	$649$
(B)	$98$	$450$
(C)	$110$	$590$
(D)	$105$	$520$
(E)	$98$	$490$
(F)	$122$	$680$

🔗

4.

Student	Correct answers on math quiz	Quiz grade
(A)	$13$	$85$
(B)	$15$	$89$
(C)	$10$	$79$
(D)	$12$	$82$
(E)	$16$	$91$
(F)	$18$	$95$

🔗

5.

The total number of barrels of oil pumped by the AQ oil company is given by the formula

🔗

N (t) = 2000 + 500 t

where

N

is the number of barrels of oil

t

days after a new well is opened. Evaluate

N (10)

and explain what it means.

🔗

6.

The number of hours required for a boat to travel upstream between two cities is given by the formula

🔗

H (v) = \frac{24}{v - 8}

where

v

represents the boat’s top speed in miles per hour. Evaluate

H (16)

and explain what it means.

🔗

Exercise Group.

For Problems 7-10, evaluate the function for the given values.

🔗

7.

F (t) = \sqrt{1 + 4 t^{2}},

F (0)

and

F (- 3)

🔗

8.

G (x) = \sqrt[3]{x - 8},

G (0)

and

G (20)

🔗

9.

h (v) = 6 - | 4 - 2 v |,

h (8)

and

h (- 8)

🔗

10.

m (p) = \frac{120}{p + 15},

m (5)

and

m (- 40)

🔗

11.

P (x) = x^{2} - 6 x + 5

🔗

Compute $P (0) .$
Find all values of $x$ for which $P (x) = 0 .$

🔗

12.

R (x) = \sqrt{4 - x^{2}}

🔗

Compute $R (0) .$
Find all values of $x$ for which $R (x) = 0 .$

🔗

Exercise Group.

For Problems 13 and 14, refer to the graphs to answer the questions.

🔗

13.

Find $f (- 2)$ and $f (2) .$
For what value(s) of $t$ is $f (t) = 4 ?$
Find the $t$ - and $f (t)$ -intercepts of the graph.
What is the maximum value of $f ?$ For what value(s) of $t$ does $f$ take on its maximum value?

🔗

🔗

14.

Find $P (- 3)$ and $P (3) .$
For what value(s) of $z$ is $P (z) = 2 ?$
Find the $z$ - and $P (z)$ -intercepts of the graph.
What is the minimum value of $P ?$ For what value(s) of $z$ does $P$ take on its minimum value?

🔗

Exercise Group.

Which of the graphs in Problems 15–18 represent functions?

🔗

Exercise Group.

For Problems 19–22, graph the function by hand.

🔗

19.

f (t) = - 2 t + 4

🔗

20.

g (s) = \frac{- 2}{3} s - 2

🔗

21.

p (x) = 9 - x^{2}

🔗

22.

q (x) = x^{2} - 16

🔗

Exercise Group.

For Problems 23–26, graph the given function on a graphing calculator. Then use the graph to solve the equations and inequalities. Round your answers to one decimal place if necessary.

🔗

23.

y = \sqrt[3]{x}

🔗

Solve $\sqrt[3]{x} = 0.8$
Solve $\sqrt[3]{x} = 1.5$
Solve $\sqrt[3]{x} > 1.7$
Solve $\sqrt[3]{x} \leq 1.26$

🔗

24.

y = \frac{1}{x}

🔗

Solve $\frac{1}{x} = 2.5$
Solve $\frac{1}{x} = 0.3125$
Solve $\frac{1}{x} \geq 0. \overset{―}{2}$
Solve $\frac{1}{x} < 5$

🔗

25.

y = \frac{1}{x^{2}}

🔗

Solve $\frac{1}{x^{2}} = 0.03$
Solve $\frac{1}{x^{2}} = 6.25$
Solve $\frac{1}{x^{2}} > 0.16$
Solve $\frac{1}{x^{2}} \leq 4$

🔗

26.

y = \sqrt{x}

🔗

Solve $\sqrt{x} = 0.707$
Solve $\sqrt{x} = 1.7$
Solve $\sqrt{x} < 1.5$
Solve $\sqrt{x} \geq 1.3$

🔗

Exercise Group.

In Problems 27–30,

y

varies directly or inversely with a power of

x .

Find the power of

x

and the constant of variation,

k .

Write a formula for each function of the form

y = k x^{n}

y = \frac{k}{x^{n}} .

🔗

27.

$x$	$y$
$2$	$4.8$
$5$	$30.0$
$8$	$76.8$
$11$	$145.2$

🔗

28.

$x$	$y$
$1.4$	$75.6$
$2.3$	$124.2$
$5.9$	$318.6$
$8.3$	$448.2$

🔗

29.

$x$	$y$
$0.5$	$40.0$
$2.0$	$10.0$
$4.0$	$5.0$
$8.0$	$2.5$

🔗

30.

$x$	$y$
$1.5$	$320.0$
$2.5$	$115.2$
$4.0$	$45.0$
$6.0$	$20.0$

🔗

31.

The distance s a pebble falls through a thick liquid varies directly with the square of the length of time

t

it falls.

🔗

If the pebble falls 28 centimeters in 4 seconds, express the distance it will fall as a function of time.
Find the distance the pebble will fall in $6$ seconds.

🔗

32.

The volume,

V,

of a gas varies directly with the temperature,

T,

and inversely with the pressure,

P,

of the gas.

🔗

If $V = 40$ when $T = 300$ and $P = 30,$ express the volume of the gas as a function of the temperature and pressure of the gas.
Find the volume when $T = 320$ and $P = 40 .$

🔗

33.

The demand for bottled water is inversely proportional to the price per bottle. If Droplets can sell 600 bottles at $8 each, how many bottles can the company sell at $10 each?

🔗

34.

The intensity of illumination from a light source varies inversely with the square of the distance from the source. If a reading lamp has an intensity of 100 lumens at a distance of 3 feet, what is its intensity 8 feet away?

🔗

35.

A person’s weight,

w,

varies inversely with the square of his or her distance,

r,

from the center of the Earth.

🔗

Express $w$ as a function of $r .$ Let $k$ stand for the constant of variation.
Make a rough graph of your function.
How far from the center of the Earth must Neil be in order to weigh one-third of his weight on the surface? The radius of the Earth is about 3960 miles.

🔗

36.

The period,

T,

of a pendulum varies directly with the square root of its length,

L .

🔗

Express $T$ as a function of $L .$ Let $k$ stand for the constant of variation.
Make a rough graph of your function.
If a certain pendulum is replaced by a new one four-fifths as long as the old one, what happens to the period?

🔗

Exercise Group.

For Problems 37 and 38, sketch a graph to illustrate the situations.

🔗

37.

Inga runs hot water into the bathtub until it is about half full. Because the water is too hot, she lets it sit for a while before getting into the tub. After several minutes of bathing, she gets out and drains the tub. Graph the water level in the bathtub as a function of time, from the moment Inga starts filling the tub until it is drained.

🔗

38.

David turns on the oven and it heats up steadily until the proper baking temperature is reached. The oven maintains that temperature during the time David bakes a pot roast. When he turns the oven off, David leaves the oven door open for a few minutes, and the temperature drops fairly rapidly during that time. After David closes the door, the temperature continues to drop, but at a much slower rate. Graph the temperature of the oven as a function of time, from the moment David first turns on the oven until shortly after David closes the door when the oven is cooling.

🔗